# Properties

 Modulus $161$ Structure $$C_{2}\times C_{66}$$ Order $132$

# Learn more

Show commands: PariGP / SageMath

sage: H = DirichletGroup(161)

pari: g = idealstar(,161,2)

## Character group

 sage: G.order()  pari: g.no Order = 132 sage: H.invariants()  pari: g.cyc Structure = $$C_{2}\times C_{66}$$ sage: H.gens()  pari: g.gen Generators = $\chi_{161}(24,\cdot)$, $\chi_{161}(120,\cdot)$

## First 32 of 132 characters

Each row describes a character. When available, the columns show the orbit label, order of the character, whether the character is primitive, and several values of the character.

Character Orbit Order Primitive $$-1$$ $$1$$ $$2$$ $$3$$ $$4$$ $$5$$ $$6$$ $$8$$ $$9$$ $$10$$ $$11$$ $$12$$
$$\chi_{161}(1,\cdot)$$ 161.a 1 no $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$
$$\chi_{161}(2,\cdot)$$ 161.m 33 yes $$1$$ $$1$$ $$e\left(\frac{28}{33}\right)$$ $$e\left(\frac{26}{33}\right)$$ $$e\left(\frac{23}{33}\right)$$ $$e\left(\frac{25}{33}\right)$$ $$e\left(\frac{7}{11}\right)$$ $$e\left(\frac{6}{11}\right)$$ $$e\left(\frac{19}{33}\right)$$ $$e\left(\frac{20}{33}\right)$$ $$e\left(\frac{5}{33}\right)$$ $$e\left(\frac{16}{33}\right)$$
$$\chi_{161}(3,\cdot)$$ 161.n 66 yes $$-1$$ $$1$$ $$e\left(\frac{26}{33}\right)$$ $$e\left(\frac{53}{66}\right)$$ $$e\left(\frac{19}{33}\right)$$ $$e\left(\frac{37}{66}\right)$$ $$e\left(\frac{13}{22}\right)$$ $$e\left(\frac{4}{11}\right)$$ $$e\left(\frac{20}{33}\right)$$ $$e\left(\frac{23}{66}\right)$$ $$e\left(\frac{7}{33}\right)$$ $$e\left(\frac{25}{66}\right)$$
$$\chi_{161}(4,\cdot)$$ 161.m 33 yes $$1$$ $$1$$ $$e\left(\frac{23}{33}\right)$$ $$e\left(\frac{19}{33}\right)$$ $$e\left(\frac{13}{33}\right)$$ $$e\left(\frac{17}{33}\right)$$ $$e\left(\frac{3}{11}\right)$$ $$e\left(\frac{1}{11}\right)$$ $$e\left(\frac{5}{33}\right)$$ $$e\left(\frac{7}{33}\right)$$ $$e\left(\frac{10}{33}\right)$$ $$e\left(\frac{32}{33}\right)$$
$$\chi_{161}(5,\cdot)$$ 161.o 66 yes $$1$$ $$1$$ $$e\left(\frac{25}{33}\right)$$ $$e\left(\frac{37}{66}\right)$$ $$e\left(\frac{17}{33}\right)$$ $$e\left(\frac{7}{33}\right)$$ $$e\left(\frac{7}{22}\right)$$ $$e\left(\frac{3}{11}\right)$$ $$e\left(\frac{4}{33}\right)$$ $$e\left(\frac{32}{33}\right)$$ $$e\left(\frac{49}{66}\right)$$ $$e\left(\frac{5}{66}\right)$$
$$\chi_{161}(6,\cdot)$$ 161.l 22 yes $$-1$$ $$1$$ $$e\left(\frac{7}{11}\right)$$ $$e\left(\frac{13}{22}\right)$$ $$e\left(\frac{3}{11}\right)$$ $$e\left(\frac{7}{22}\right)$$ $$e\left(\frac{5}{22}\right)$$ $$e\left(\frac{10}{11}\right)$$ $$e\left(\frac{2}{11}\right)$$ $$e\left(\frac{21}{22}\right)$$ $$e\left(\frac{4}{11}\right)$$ $$e\left(\frac{19}{22}\right)$$
$$\chi_{161}(8,\cdot)$$ 161.i 11 no $$1$$ $$1$$ $$e\left(\frac{6}{11}\right)$$ $$e\left(\frac{4}{11}\right)$$ $$e\left(\frac{1}{11}\right)$$ $$e\left(\frac{3}{11}\right)$$ $$e\left(\frac{10}{11}\right)$$ $$e\left(\frac{7}{11}\right)$$ $$e\left(\frac{8}{11}\right)$$ $$e\left(\frac{9}{11}\right)$$ $$e\left(\frac{5}{11}\right)$$ $$e\left(\frac{5}{11}\right)$$
$$\chi_{161}(9,\cdot)$$ 161.m 33 yes $$1$$ $$1$$ $$e\left(\frac{19}{33}\right)$$ $$e\left(\frac{20}{33}\right)$$ $$e\left(\frac{5}{33}\right)$$ $$e\left(\frac{4}{33}\right)$$ $$e\left(\frac{2}{11}\right)$$ $$e\left(\frac{8}{11}\right)$$ $$e\left(\frac{7}{33}\right)$$ $$e\left(\frac{23}{33}\right)$$ $$e\left(\frac{14}{33}\right)$$ $$e\left(\frac{25}{33}\right)$$
$$\chi_{161}(10,\cdot)$$ 161.o 66 yes $$1$$ $$1$$ $$e\left(\frac{20}{33}\right)$$ $$e\left(\frac{23}{66}\right)$$ $$e\left(\frac{7}{33}\right)$$ $$e\left(\frac{32}{33}\right)$$ $$e\left(\frac{21}{22}\right)$$ $$e\left(\frac{9}{11}\right)$$ $$e\left(\frac{23}{33}\right)$$ $$e\left(\frac{19}{33}\right)$$ $$e\left(\frac{59}{66}\right)$$ $$e\left(\frac{37}{66}\right)$$
$$\chi_{161}(11,\cdot)$$ 161.p 66 yes $$-1$$ $$1$$ $$e\left(\frac{5}{33}\right)$$ $$e\left(\frac{7}{33}\right)$$ $$e\left(\frac{10}{33}\right)$$ $$e\left(\frac{49}{66}\right)$$ $$e\left(\frac{4}{11}\right)$$ $$e\left(\frac{5}{11}\right)$$ $$e\left(\frac{14}{33}\right)$$ $$e\left(\frac{59}{66}\right)$$ $$e\left(\frac{23}{66}\right)$$ $$e\left(\frac{17}{33}\right)$$
$$\chi_{161}(12,\cdot)$$ 161.n 66 yes $$-1$$ $$1$$ $$e\left(\frac{16}{33}\right)$$ $$e\left(\frac{25}{66}\right)$$ $$e\left(\frac{32}{33}\right)$$ $$e\left(\frac{5}{66}\right)$$ $$e\left(\frac{19}{22}\right)$$ $$e\left(\frac{5}{11}\right)$$ $$e\left(\frac{25}{33}\right)$$ $$e\left(\frac{37}{66}\right)$$ $$e\left(\frac{17}{33}\right)$$ $$e\left(\frac{23}{66}\right)$$
$$\chi_{161}(13,\cdot)$$ 161.l 22 yes $$-1$$ $$1$$ $$e\left(\frac{3}{11}\right)$$ $$e\left(\frac{15}{22}\right)$$ $$e\left(\frac{6}{11}\right)$$ $$e\left(\frac{3}{22}\right)$$ $$e\left(\frac{21}{22}\right)$$ $$e\left(\frac{9}{11}\right)$$ $$e\left(\frac{4}{11}\right)$$ $$e\left(\frac{9}{22}\right)$$ $$e\left(\frac{8}{11}\right)$$ $$e\left(\frac{5}{22}\right)$$
$$\chi_{161}(15,\cdot)$$ 161.j 22 no $$-1$$ $$1$$ $$e\left(\frac{6}{11}\right)$$ $$e\left(\frac{4}{11}\right)$$ $$e\left(\frac{1}{11}\right)$$ $$e\left(\frac{17}{22}\right)$$ $$e\left(\frac{10}{11}\right)$$ $$e\left(\frac{7}{11}\right)$$ $$e\left(\frac{8}{11}\right)$$ $$e\left(\frac{7}{22}\right)$$ $$e\left(\frac{21}{22}\right)$$ $$e\left(\frac{5}{11}\right)$$
$$\chi_{161}(16,\cdot)$$ 161.m 33 yes $$1$$ $$1$$ $$e\left(\frac{13}{33}\right)$$ $$e\left(\frac{5}{33}\right)$$ $$e\left(\frac{26}{33}\right)$$ $$e\left(\frac{1}{33}\right)$$ $$e\left(\frac{6}{11}\right)$$ $$e\left(\frac{2}{11}\right)$$ $$e\left(\frac{10}{33}\right)$$ $$e\left(\frac{14}{33}\right)$$ $$e\left(\frac{20}{33}\right)$$ $$e\left(\frac{31}{33}\right)$$
$$\chi_{161}(17,\cdot)$$ 161.o 66 yes $$1$$ $$1$$ $$e\left(\frac{32}{33}\right)$$ $$e\left(\frac{17}{66}\right)$$ $$e\left(\frac{31}{33}\right)$$ $$e\left(\frac{5}{33}\right)$$ $$e\left(\frac{5}{22}\right)$$ $$e\left(\frac{10}{11}\right)$$ $$e\left(\frac{17}{33}\right)$$ $$e\left(\frac{4}{33}\right)$$ $$e\left(\frac{35}{66}\right)$$ $$e\left(\frac{13}{66}\right)$$
$$\chi_{161}(18,\cdot)$$ 161.m 33 yes $$1$$ $$1$$ $$e\left(\frac{14}{33}\right)$$ $$e\left(\frac{13}{33}\right)$$ $$e\left(\frac{28}{33}\right)$$ $$e\left(\frac{29}{33}\right)$$ $$e\left(\frac{9}{11}\right)$$ $$e\left(\frac{3}{11}\right)$$ $$e\left(\frac{26}{33}\right)$$ $$e\left(\frac{10}{33}\right)$$ $$e\left(\frac{19}{33}\right)$$ $$e\left(\frac{8}{33}\right)$$
$$\chi_{161}(19,\cdot)$$ 161.o 66 yes $$1$$ $$1$$ $$e\left(\frac{1}{33}\right)$$ $$e\left(\frac{49}{66}\right)$$ $$e\left(\frac{2}{33}\right)$$ $$e\left(\frac{28}{33}\right)$$ $$e\left(\frac{17}{22}\right)$$ $$e\left(\frac{1}{11}\right)$$ $$e\left(\frac{16}{33}\right)$$ $$e\left(\frac{29}{33}\right)$$ $$e\left(\frac{31}{66}\right)$$ $$e\left(\frac{53}{66}\right)$$
$$\chi_{161}(20,\cdot)$$ 161.k 22 yes $$1$$ $$1$$ $$e\left(\frac{5}{11}\right)$$ $$e\left(\frac{3}{22}\right)$$ $$e\left(\frac{10}{11}\right)$$ $$e\left(\frac{8}{11}\right)$$ $$e\left(\frac{13}{22}\right)$$ $$e\left(\frac{4}{11}\right)$$ $$e\left(\frac{3}{11}\right)$$ $$e\left(\frac{2}{11}\right)$$ $$e\left(\frac{1}{22}\right)$$ $$e\left(\frac{1}{22}\right)$$
$$\chi_{161}(22,\cdot)$$ 161.d 2 no $$-1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$-1$$ $$1$$ $$1$$ $$1$$ $$-1$$ $$-1$$ $$1$$
$$\chi_{161}(24,\cdot)$$ 161.h 6 no $$-1$$ $$1$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$-1$$ $$1$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{5}{6}\right)$$
$$\chi_{161}(25,\cdot)$$ 161.m 33 yes $$1$$ $$1$$ $$e\left(\frac{17}{33}\right)$$ $$e\left(\frac{4}{33}\right)$$ $$e\left(\frac{1}{33}\right)$$ $$e\left(\frac{14}{33}\right)$$ $$e\left(\frac{7}{11}\right)$$ $$e\left(\frac{6}{11}\right)$$ $$e\left(\frac{8}{33}\right)$$ $$e\left(\frac{31}{33}\right)$$ $$e\left(\frac{16}{33}\right)$$ $$e\left(\frac{5}{33}\right)$$
$$\chi_{161}(26,\cdot)$$ 161.n 66 yes $$-1$$ $$1$$ $$e\left(\frac{4}{33}\right)$$ $$e\left(\frac{31}{66}\right)$$ $$e\left(\frac{8}{33}\right)$$ $$e\left(\frac{59}{66}\right)$$ $$e\left(\frac{13}{22}\right)$$ $$e\left(\frac{4}{11}\right)$$ $$e\left(\frac{31}{33}\right)$$ $$e\left(\frac{1}{66}\right)$$ $$e\left(\frac{29}{33}\right)$$ $$e\left(\frac{47}{66}\right)$$
$$\chi_{161}(27,\cdot)$$ 161.l 22 yes $$-1$$ $$1$$ $$e\left(\frac{4}{11}\right)$$ $$e\left(\frac{9}{22}\right)$$ $$e\left(\frac{8}{11}\right)$$ $$e\left(\frac{15}{22}\right)$$ $$e\left(\frac{17}{22}\right)$$ $$e\left(\frac{1}{11}\right)$$ $$e\left(\frac{9}{11}\right)$$ $$e\left(\frac{1}{22}\right)$$ $$e\left(\frac{7}{11}\right)$$ $$e\left(\frac{3}{22}\right)$$
$$\chi_{161}(29,\cdot)$$ 161.i 11 no $$1$$ $$1$$ $$e\left(\frac{7}{11}\right)$$ $$e\left(\frac{1}{11}\right)$$ $$e\left(\frac{3}{11}\right)$$ $$e\left(\frac{9}{11}\right)$$ $$e\left(\frac{8}{11}\right)$$ $$e\left(\frac{10}{11}\right)$$ $$e\left(\frac{2}{11}\right)$$ $$e\left(\frac{5}{11}\right)$$ $$e\left(\frac{4}{11}\right)$$ $$e\left(\frac{4}{11}\right)$$
$$\chi_{161}(30,\cdot)$$ 161.p 66 yes $$-1$$ $$1$$ $$e\left(\frac{13}{33}\right)$$ $$e\left(\frac{5}{33}\right)$$ $$e\left(\frac{26}{33}\right)$$ $$e\left(\frac{35}{66}\right)$$ $$e\left(\frac{6}{11}\right)$$ $$e\left(\frac{2}{11}\right)$$ $$e\left(\frac{10}{33}\right)$$ $$e\left(\frac{61}{66}\right)$$ $$e\left(\frac{7}{66}\right)$$ $$e\left(\frac{31}{33}\right)$$
$$\chi_{161}(31,\cdot)$$ 161.n 66 yes $$-1$$ $$1$$ $$e\left(\frac{29}{33}\right)$$ $$e\left(\frac{35}{66}\right)$$ $$e\left(\frac{25}{33}\right)$$ $$e\left(\frac{7}{66}\right)$$ $$e\left(\frac{9}{22}\right)$$ $$e\left(\frac{7}{11}\right)$$ $$e\left(\frac{2}{33}\right)$$ $$e\left(\frac{65}{66}\right)$$ $$e\left(\frac{4}{33}\right)$$ $$e\left(\frac{19}{66}\right)$$
$$\chi_{161}(32,\cdot)$$ 161.m 33 yes $$1$$ $$1$$ $$e\left(\frac{8}{33}\right)$$ $$e\left(\frac{31}{33}\right)$$ $$e\left(\frac{16}{33}\right)$$ $$e\left(\frac{26}{33}\right)$$ $$e\left(\frac{2}{11}\right)$$ $$e\left(\frac{8}{11}\right)$$ $$e\left(\frac{29}{33}\right)$$ $$e\left(\frac{1}{33}\right)$$ $$e\left(\frac{25}{33}\right)$$ $$e\left(\frac{14}{33}\right)$$
$$\chi_{161}(33,\cdot)$$ 161.o 66 yes $$1$$ $$1$$ $$e\left(\frac{31}{33}\right)$$ $$e\left(\frac{1}{66}\right)$$ $$e\left(\frac{29}{33}\right)$$ $$e\left(\frac{10}{33}\right)$$ $$e\left(\frac{21}{22}\right)$$ $$e\left(\frac{9}{11}\right)$$ $$e\left(\frac{1}{33}\right)$$ $$e\left(\frac{8}{33}\right)$$ $$e\left(\frac{37}{66}\right)$$ $$e\left(\frac{59}{66}\right)$$
$$\chi_{161}(34,\cdot)$$ 161.k 22 yes $$1$$ $$1$$ $$e\left(\frac{9}{11}\right)$$ $$e\left(\frac{1}{22}\right)$$ $$e\left(\frac{7}{11}\right)$$ $$e\left(\frac{10}{11}\right)$$ $$e\left(\frac{19}{22}\right)$$ $$e\left(\frac{5}{11}\right)$$ $$e\left(\frac{1}{11}\right)$$ $$e\left(\frac{8}{11}\right)$$ $$e\left(\frac{15}{22}\right)$$ $$e\left(\frac{15}{22}\right)$$
$$\chi_{161}(36,\cdot)$$ 161.i 11 no $$1$$ $$1$$ $$e\left(\frac{3}{11}\right)$$ $$e\left(\frac{2}{11}\right)$$ $$e\left(\frac{6}{11}\right)$$ $$e\left(\frac{7}{11}\right)$$ $$e\left(\frac{5}{11}\right)$$ $$e\left(\frac{9}{11}\right)$$ $$e\left(\frac{4}{11}\right)$$ $$e\left(\frac{10}{11}\right)$$ $$e\left(\frac{8}{11}\right)$$ $$e\left(\frac{8}{11}\right)$$
$$\chi_{161}(37,\cdot)$$ 161.p 66 yes $$-1$$ $$1$$ $$e\left(\frac{19}{33}\right)$$ $$e\left(\frac{20}{33}\right)$$ $$e\left(\frac{5}{33}\right)$$ $$e\left(\frac{41}{66}\right)$$ $$e\left(\frac{2}{11}\right)$$ $$e\left(\frac{8}{11}\right)$$ $$e\left(\frac{7}{33}\right)$$ $$e\left(\frac{13}{66}\right)$$ $$e\left(\frac{61}{66}\right)$$ $$e\left(\frac{25}{33}\right)$$
$$\chi_{161}(38,\cdot)$$ 161.o 66 yes $$1$$ $$1$$ $$e\left(\frac{29}{33}\right)$$ $$e\left(\frac{35}{66}\right)$$ $$e\left(\frac{25}{33}\right)$$ $$e\left(\frac{20}{33}\right)$$ $$e\left(\frac{9}{22}\right)$$ $$e\left(\frac{7}{11}\right)$$ $$e\left(\frac{2}{33}\right)$$ $$e\left(\frac{16}{33}\right)$$ $$e\left(\frac{41}{66}\right)$$ $$e\left(\frac{19}{66}\right)$$
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